Metatheory - University of Cambridge

5. Soundness 38
Again, it is easy to check that line 4 is shiny. As is every line in this TFL-proof.
I want to show that this is no coincidence. That is, I want to prove:
Lemma 5.2. Every line of every TFL-proof is shiny.
Then we will know that we have never gone astray, on any line of a proof.
Indeed, given Lemma 5.2, it will be easy to prove the Soundness Theorem:
Proof of Soundness Theorem, given Lemma 5.2. Suppose Γ ⊢ C . Then there
is a TFL-proof, with C appearing on its last line, whose only undischarged
assumptions are among Γ. Lemma 5.2 tells us that every line on every TFLproof
is shiny. So this last line is shiny, i.e. Γ ⊨ C .
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It remains to prove Lemma 5.2.
To do this, we observe that every line of any TFL-proof is obtained by
applying some rule. So what I want to show is that no application of a rule
of TFL’s proof system will lead us astray. More precisely, say that a rule of
inference is rule-sound iff for all TFL-proofs, if we obtain a line on a TFLproof
by applying that rule, and every earlier line in the TFL-proof is shiny,
then our new line is also shiny. What I need to show is that every rule in TFL’s
proof system is rule-sound.
I shall do this in the next section. But having demonstrated the rulesoundness
of every rule, Lemma 5.2 will follow immediately:
Proof of Lemma 5.2, given that every rule is rule-sound. Fix any line, line n,
on any TFL-proof. The sentence written on line n must be obtained using a
formal inference rule which is rule-sound. This is to say that, if every earlier
line is shiny, then line n itself is shiny. Hence, by strong induction on the length
of TFL-proofs, every line of every TFL-proof is shiny.
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Note that this proof appeals to a principle of strong induction on the length of
TFL-proofs. This is the first time we have seen that principle, and you should
pause to confirm that it is, indeed, justified. (Hint: compare the justification
for the Principle of Strong Induction on Length of sentences from §1.4.)
5.2 Checking each rule
It remains to show that every rule is rule-sound. This is actually much easier
than many of the proofs in earlier chapters. But it is also time-consuming, since
we need to check each rule individually, and TFL’s proof system has plenty
of rules! To speed up the process marginally, I shall introduce a convenient
abbreviation: ‘∆ i ’ will abbreviate the assumptions (if any) on which line i
depends in our TFL-proof (context will indicate which TFL-proof I have in
mind).
Lemma 5.3. Introducing an assumption is rule-sound.
Proof. If A is introduced as an assumption on line n, then A is among ∆ n ,
and so ∆ n ⊨ A.
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Lemma 5.4. ∧I is rule-sound.